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Information Circular 8873 

Computer Simulation Applied 
to the Separation of Porous Leach 
Residue Solids From Liquor 
by Horizontal Belt Filtration 

By Daniel T. Rogers and Roy T. Sorensen, Jr. 




UNITED STATES DEPARTMENT OF THE INTERIOR 
James G. Watt, Secretary 

BUREAU OF MINES 
Robert C. Horton, Director 



IC 8873 









Bureau of Mines Information Circular/1982 



Computer Simulation Applied 
to the Separation of Porous Leach 
Residue Solids From Liquor 
by Horizontal Belt Filtration 




By Daniel T. Rogers and Roy T. Sorensen, Jr. 




UNITED STATES DEPARTMENT OF THE INTERIOR 



^ 



;w 



^f\0 



i*^ 



As the Nation's principal conservation agency, the Department of the 
Interior has responsibility for most of our nationally owned public lands 
and natural resources. This includes fostering the wisest use of our land 
and water resources, protecting our fish and wildlife, preserving the 
environmental and cultural values of our national parks and historical 
places, and providing for the enjoyment of life through outdoor recreation. 
The Department assesses our energy and mineral resources and works to 
assure that their development is in the best interests of all our people. The 
Department also has a major responsibility for American Indian reserva- 
tion communities and for people who live in island territories under U.S. 
administration. 




This publication has been cataloged as follows 



Rogers, Daniel T 

Computer simulation applied to the separation of porous 
leach residue solids from liquor by horizontal belt filtration. 

(Information circular / Bureau of Mines ; 3873) 
Supt. of Docs, no.: I 28.27:8873- 

1. Aluminum— Metallurgy— Mathematical models. 2. Aluminum- 
Metallurgy— Data processing. 3. JFilters and filtration— Mathematical 
models. 4. Filters and filtration— Data processing. 5. Aluminum 
oxide. 6. Hydrochloric acid. I. Sorensen, Roy T. II. Title. HI. 
Series: Information circular (United States. Bureau of Mines) ; 8873. 

TN295.U4 [TN775] 622s [669'.722] 81-607166 AACR2 



For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C, 20402 



Ill 



CONTENTS 



V) 



Page 

Abstract 1 

Introduction 1 

Belt filtration modeling 2 

Background 2 

Filtration and washing 2 

Process description 2 

Definition of washing efficiency 2 

Prior material balance calculations 2 

Prior horizontal belt filter modeling 3 

Selection of a model 3 

Purpose of model development 3 

Ground rules for model development 3 

The existing perfect-mixing-cells-in-series 

(PMCS) model 3 

Description 3 

Discussion 3 

The diffusion model 3 

Description 3 

Discussion 3 

The shrinking voids model 3 

Description 3 

Discussion 3 

Derivation of equations for a single wash- 
filtration stage 4 

Fundamental PMCS equations 4 

Development of diffusion and shrinking voids 

model equations 4 

Balance around the wash step 4 

Diffusion model: migration of AI2O3 from 

cake voids 5 



Page 

Shrinking voids model: hypothetical voids 

shrinkage 6 

Completion of countercurrent calculations . . 7 
Equations for liquor weight and volume 

balance 7 

Summary of material balance procedure 7 

Sample calculations 9 

First material balance trial 9 

Procedure following first balance trial 9 

Evaluation by least squares error 9 

Goodness of fit 9 

Least squares fit 10 

Model evaluation using miniplant data 10 

Application of the shrinking voids model 12 

Development of material balances from plant 

data 12 

Prediction of hypothetical plant material 

balances 12 

Optimization of horizontal belt 

filtration operations 12 

Summary and conclusions 13 

Appendix A.— Material balance by Fortran 

computer 14 

Appendix B.— Material balance using a 

programmable calculator 21 

Appendix C— Measurement of leach residue 

porosities and densities 25 

Appendix D.— Cake liquor density and volume 

determinations 26 

Appendix E.— Nomenclature 27 



ILLUSTRATIONS 

1. Schematic of continuous horizontal belt filtration circuit >. 2 

2: Schematic for two mathematical molds (diffusion and shrinking voids) of the displace 

washing of the continuous belt filtration 4 

3. Summary of equations used in calculating a material balance around a single wash step 5 

4. Cake liquor solute concentration, Cf, as a function of diffusion time 6 

5. Sample filtration circuit material balance form 8 

B-1. Schematic of TI-59 program for material balance using the shrinking voids model balance 24 



TABLES 

1. Miniplant belt filtration data, test 1-3 7 

2. Summary of a material balance calculation method, test 1-3 10 

3. Summary of sums of squares of errors (SSE's) for test 1-3 material balance 10 

4. Material balances for seven miniplant tests 11 

5. Comparison of best-fit material balance with shrinking voids model material balance 11 

6. Average filtration parameters for miniplant leach residues containing 123 lb of dry solids 12 

7. Predicted AI2O3 filtration losses under various operating configurations 12 

8. Predicted form filtrate concentrations under various operating configurations 13 

C-1. Summary of 1-day-6ld leach residue densities and porosities 25 



COMPUTERSIMULATIONAPPLIEDTOTHESEPARATIONOFPOROUS LEACH 
RESIDUE SOLIDS FROM LIQUOR BY HORIZONTAL BELT FILTRATION 



By Daniel T. Rogers 1 and Roy T. Sorenson, Jr. 



ABSTRACT 

The Bureau of Mines, in its alumina miniplant project to investigate alumina recovery 
from domestic, nonbauxitic ores, has conducted research on the use of a hydrochloric 
acid leaching, gas sparging crystallization technology. An important element of this 
research is the efficient separation of undissolved, siliceous residue from AICI 3 -bearing 
leach liquors by continuous, horizontal, countercurrent vacuum belt filtration. 

In an effort to calculate material balances quickly and to predict material balances 
based on different belt filtration configurations, the perfect-mixing-cells-in-series model 
(PMCS) for calculating material balances around belt filters was used. Because of the 
porous nature of the solids, the model produced erroneous material balances. Therefore 
a reliable model, the shrinking voids model, was developed postulating the presence of 
an unwashable voids liquor volume that decreases with decreasing liquor AICI 3 concen- 
tration. This volume decrease postulation is equivalent to assuming that dilute liquors 
flow more freely, causing more voids liquor volume to become washable. 

Least-squares based computer programs are provided, which are useful not only in 
producing material balances from plant data but also in predicting balances for 
untested configurations using the same feed materials. 



INTRODUCTION 

As the United States is dependent on foreign sources of bauxite for the alumina re- 
quired for aluminum production, domestic kaolinitic clay, hereafter referred to as clay, 
is being explored as an alternate source of aluminum. The Bureau of Mines is presently 
testing technology in which aqueous hydrochloric acid (HCI) is used to leach aluminum 
from calcined clay. This leaching produces an aqueous aluminum chloride (AICI 3 ) solu- 
tion that must be separated from the siliceous solids not dissolved by HCI. Two possi- 
ble methods for achieving this separation, vacuum filtration and classifier-thickener 
systems, were tested. 

Because of the time required to calculate material balances from the filtration data 
obtained in the alumina miniplant operation, and the need to predict material balances 
for different possible belt filtration configurations, an existing computer model, the 
perfect-mixing-cells-in-series model (PMCS), was used for these calculations. As this 
model did not result in accurate calculations because of the porous nature of the solids, 
a new model, the shrinking voids model, was developed to achieve the following 
objectives: 

1. Calculate material balances rapidly from alumina miniplant data. 

2. Predict material balances for belt filtration configurations different from those 
tested. 

3. Predict filtration balances for slurries containing porous solids, when the existing 
model is not expected to be usable. 

'Chemist. 
'Metallurgist. 
Both authors are with the Boulder City Engineering Laboratory, Bureau of Mines, Boulder City, Nev. 




BELT FILTRATION MODELING 



BACKGROUND 

During investigations in the Bureau's alumina miniplant 
project for extracting alumina from clay by a hydrochloric 
acid leaching technology, separation of the undissolved, 
siliceous residue from the AICl3-bearing solution was 
found to be a major problem. The filtration was slow and 
the washing efficiencies were poor. Two systems for 
solids-liquid separation, a clast-ifier-thickener system and 
a belt filtration system, were investigated. The belt filtra- 
tion system has proven to be a satisfactory approach to 
the solids-liquid separation problem. 

In an effort to calculate accurate material balances 
quickly around the belt filtration system and to predict 
results from various belt filtration configurations, it was 
necessary to devise a new modeling technique. 



Filtration and Washing 
Process Description 

The method of solids-liquid separation covered in this 
report is horizontal vacuum belt filtration. The alumina 
miniplant filter consists of a moving belt onto which slurry 
(with or without the addition of flocculant solution) is 
deposited under the impetus of a vacuum. This deposition 
process separates the alumina-bearing liquor from the 
cake that the belt carries through one or more sprays 
(washes) for recovery of additional alumina values. 

The filtrate at each stage is then transferred countercur- 
rent to the direction of cake movement for use as a wash 
spray (see fig. 1). Countercurrent filtration and washing is 
generally considered the most efficient method for max- 
imum product recovery for a given amount of wash water 
added in the final cake wash. 



Definition of Washing Efficiency 

Ideally, after the cake is washed in the filtration pro- 
cess, the wash liquor will have displaced all of the more 
concentrated solute in the liquor with wash liquor to give 
perfect washing. However, since perfect washing is sel- 
dom achieved, a measure of washing efficiency is needed 
to evaluate the process. The R value (equal to 100 minus 
the efficiency) measures the percent of solute remaining 
in the cake after washing, and subtracts any solute added 
in the wash. 3 



R = 



c 2- C v 
Ci - C u 



(100), 



(1) 



where C2 = solute concentration (pounds per gallon) in 

washed cake liquor, 
C1 = solute concentration (pounds per gallon) in 

feed cake liquor, 
and C w = solute concentration (pounds per gallon) in 

wash liquor. 
This R value will, since it is actually a function of the wash 
ratio, N, of wash liquor volume to cake liquor volume, 
change as N changes. 

This report will, however, refer to the residuals, R, only in 
passing because the ultimate goal is to determine solute 
losses through the final wash cake. 

Prior Material Balance Calculations 

Heretofore material balances for horizontal belt filtra- 
tion in the clay-HCI miniplant were made by laborious 



3 Dahlstrom, D. A., and Silverblatt, C. E. Continuous Vacuum and 
Pressure Filtration. Ch. in Solid/Liquid Separation Equipment 
Scaleup, ed. D. B. Purchas. Upland Press Ltd., Croydon, England, 
1977, p. 477. 




2nd wash cake 



Farm filtrate 



2nd wash filtrate 



FIGURE 1.— Schematic of continuous horizontal belt filtration circuit. 



calculations. A solids-liquid material balance was made 
from plant input flow rates and cake moisture contents. 
An alumina balance of the liquor stream was then made by 
best-fitting or equal-weighting the AI2O3 analysis of the 
liquor in the filtration test samples. 

Using these methods, material balances were made for 
seven horizontal belt filtration tests, and the balances 
were later used for evaluating the models developed in 
this report. 

Prior Horizontal Belt Filter Modeling 

Previously, in order to predict material balances for cir- 
cuits employing various wash liquor rates or number of 
washes, the following two assumptions were required: 

1 . R values for additional stages were equal to the 
average R value in the first two stages. 

2 . Cake moisture decreased in a straight line with the 
number of wash stages. 



SELECTION OF A MODEL 

Purpose of Model Development 

Miniplant filtration data are most easily applied when 
summarized in a mathematical model. Such models must, 
of course, predict stream compositions as close as possi- 
ble to those observed. The models should also predict 
reliable material balances for continuous filtration system 
configurations different from those tested in the 
miniplant. Such models facilitate selection of the most 
cost-effective filter system configuration. For example, 
both alumina losses through the final wash cake and the 
degree of product liquor dilution by wash water for any 
number of wash stages or any volume of wash water addi- 
tion are predicted rather than measured. 

Ground Rules for Model Development 

The models in this study are all guided by theoreti- 
cally logical constraints on the relationships among filtra- 
tion flow streams. These constraints are applied in such a 
manner that the final material balance stream composi- 
tions differ minimally from measured compositions— us- 
ing a least squares error procedure. The following logical 
constraints common to all these models are referred to as 
ground rules: 

1 . No solid or liquid losses occur. 

2 . The filtrates contain no solids. 

3 . Liquor volume input equals liquor volume output for 

each washing-filtration stage. 

4 . The ratio of solute in the aqueous phase to that in 

the solid phase remains constant (presumably no 
solute exists in the solid phase for the study 
material). 

5 . Total cake liquor volume is constant from one 

washing stage to the next. 

The Existing Perfect-Mixing-Cells-in-Series 
(PMCS) Model 

Description 

The PMCS model 4 uses the previously mentioned 
ground rules. It assumes that cake washing is described 




'Torniak, A. Predict Performance of Belt-Filter Washing. Chem. Eng. v. 86, 
No. 9, Apr. 23, 1979, pp. 143-146. 

Tomiak, A. Theoretical Recoveries in Filter Cake Reslurrying and 
Washing. AlChE J., v. 19, No. 1, January 1973, pp. 76-84. 



by the complete equilibration of cake liquor with wash li- 
quor as the later moves through each of a number of 
perfect-mixing cells in the cake. The number of these cells 
is an intensive property, j, which does not change with 
cake size. 

Discussion 

This model specifies that j must be an integer greater 
than zero. Testing of this model with alumina miniplant 
data reveals that unless j values are extrapolated to some 
value less than 1, the model fails completely. It fails 
because the leached solids are very porous, indicating an 
imperfect rather than a perfect mixing (PMCS model) con- 
dition. 



The Diffusion Model 



Description 



Because the PMCS model inadequately describes filtra- 
tion of the alumina miniplant solids, it was decided to 
develop a model that recognized the presence of a large 
voids volume inside the leached residue particles, which 
has been determined to be 54.6 pet (see appendix C). 

The model assumes the cake liquor to be composed of 
the following two separate fractions: 

Internal— liquor entrained in the pores of the solids. 
External— liquor outside the pores and between the 
solid particles. 

During a wash, the external fraction is washed from the 
cake solids according to the PMCS model (with j cells) and 
the internal fraction is untouched by the wash liquor. After 
excess liquor is filtered from the cake, the fractions dif- 
fuse into one another for a specified time (see fig. 2). 

Discussion 

The diffusion model is a considerable improvement over 
the PMCS model. However, it predicts that alumina resid- 
uals, R, are constant from one wash stage to the next. In 
the miniplant, however, R values actually decrease signif- 
icantly with decreasing liquor concentrations. Attempts to 
eliminate this discrepancy lead to the development of a 
better model. 



The Shrinking Voids Model 



Description 



This model assumes that washing efficiency changes 
as cake liquor concentration changes. Presumably, liquor 
viscosity decreases (fluidity increases) as the liquor 
becomes more dilute, which causes the solid particle 
voids liquors to be more readily washable (displaceable). 
This increased washing efficiency is expressed as a de- 
crease in the volume of nondisplaceable voids as the 
liquor concentration decreases (see fig. 2). 

Discussion 

This model will be shown to be the best of the three 
treated in this report. However, it has not been verified us- 
ing any other configuration than that in the miniplant two- 
stage washing and filtering system. 

It might be objected that a more realistic model would 
incorporate AICI 3 diffusion during, as well as after, 
washing. It is, however, felt that diffusion during washing 
is small, which makes the simplicity of the shrinking voids 
model very attractive. 



Wash liquor (I) 



New feed cake 
(Shrinking voids model) (4) 



Unwoshobie liquor 
in voids 



Washable liquor 
film 




Wash filtrate (2) 



New feed cake 
[Diffusion model) (4) 



FIGURE 2.— Schematic for two mathematical models (diffusion and shrinking voids) of the displace washing of acid- 
leached calcined kaolin residues during continuous belt filtration. 



DERIVATION OF EQUATIONS FOR A 
SINGLE WASH-FILTRATION STAGE 

Fundamental PMCS Equations 

The basic PMCS (with j perfect-mixing cells) equation 
describing the material balance around a single wash step 
is 

* = (N^-f^) /(N-f), (2) 

Vi \ V 2 M ) 

where N = V-|/V , the wash ratio, 

f = fraction of salt removed during a wash with 
salt-free wash liquor, 
e-jN j-1 gN)k 

and A n , V n = alumina weights and liquor volumes for the 
liquor stream, n, in figure 3 (n = o for 
feed cake, n = 1 for wash liquor, n = 2 
for wash filtrate, n = 3 for initial wash 
cake, and n = 4 for equilibrated wash 
cake). 

Subscripts e and i attached to cake liquor volumes and 

alumina weights refer to external and internal fractions 

respectively. 



Development of Diffusion and Shrinking 
Voids Model Equations 

Balance Around the Wash Step 

' Both the diffusion and shrinking voids models begin 
with the application of the PMCS equation 2. However, for 
this application, cake liquor will consist only of the 
displaceable external fraction. Therefore, the cake liquor 
alumina weight (Aq = A 0I + A^) and liquor volume (V = 
Voi + v pe) are temporarily considered equal to Aoe and V oi 
respectively. Thus, equation 2 becomes 

^1 = ^.f^Wf). (3) 



This means the wash ratio is now N = V^Noe- Also, 
since cake volumes do not (ground rule 5) change during 
washing, Vi equals V2. Therefore, the wash liquor alumina 
weight can be determined from 



A1 = 



, V n A 2 

V 1 — I • — i 

1 Voe V! 



Aoe 



/(N-f), 



(A 2 - f AoeWN-f), 



_Vl 

Voe 

N(A 2 - f Aoe)/(N-f). 



(4) 







































Feed C3ke,next step (4) 






Wash liquor (I) 






. _( A o, + A 3e) , 
A 4'>oi + V oe) V4> 

A 4e rA oi +A 3e~ A 4i 

,, -v _ K ( A oe" A 3e) 

v 4i " v oi K (V +v , 

( v oi +v oe) 

V 4e =V j +v oe -v 4j 




At=j£r<A 2 -fA oe ) 

v, =v 2 


T3 

C 

o 

c 
o 

Q 


o> 

o 

-XL 

C 
JZ 

"O 

o 

> 


























(Shrinking voids model) 






Feed cake(O) 


< ' 


Washed cake(3) 




A j = specified 
Aoe= specified 
V j = specified 
V oe = specified 


f Was 


hing \ 
d V 




A-, = A 
"3i oi 

A 3e ; V^l -A 2 

V 3l = V o, 

v 3e = V oe 




»■ 


y filtr 


jtion / 




>x 














Diffusion on 

i 








Feed cake, next step (4) 






< 


' 




-tc , -tc 
A 4i =A ,e +V 0l (l-e 

A 4e =A oi +A 3e- A 4i 
v 4, =v o, 

v 4e ^v oe 


/A ,4A 3e \ 
Woi +V e / 






Wash filtrate (2) 






Ag r estimated 
V 2 ~ specified 




















(Diffusion model) 





FIGURE 3.— Equations used in calculating a material balance around a single wash step. 



Now the internal liquor parameters, Aqj and V j, tem- 
porarily set to zero, are again permitted nonzero values. 
Then the external cake alumina weight follows from 
material balance A 3e = A i + Aoe + A-| - A 2 - A 3 j, and 
since A 3i = A oi , gives A 3e = A^ + At - A 2 - 

Diffusion Model: Migration of Al 2 3 From Cake Voids 

At this point, the external cake liquor will be highly defi- 
cient in Al 2 3 . It is, therefore, proposed that a specified 
amount of diffusion will occur to change this situation. 
The occurrence of this process can be expressed by a first 
order decay curve illustrated in figure 4. The equation is 



CfC eq = (Cj - C eq ) e tc , 



(5) 



where tc, a positive time constant, is actually the product 
of a diffusion time interval, t, and a diffusion rate constant, 
c (which must be positive). Cj, Cf, and C eq are the initial, 
final, and equilibrium alumina concentrations in the liquor. 
The equilibrium concentration is, of course, defined by 



alumina weight A^ + A^ 



C = 
eq liquor volume 



(6) 



where the subscript 3 refers to the initial product cake 
liquor. Then for the subscript 4, which refers to the final 



product cake liquor, the relationships Cj = A 3 j/V 3j , Cf = 
A4j/V 4 j, and C eq of equation 6 can be substituted into equa- 
tion 5 so that A 4i may be determined. This gives 



A4I A 3i + A^ 

V 4i V 3i + Vae 



A 3 j A 3i + A;je 

V 3j V 3i + Vae 



e tc .(7) 



Now, since cake volume does not change, the relation- 
ships V 4i = V 3i = VojandV^ = V^ = V^ must hold. Also, 
Aqj = A 3 j must hold, since no alumina has been removed 
from the particle interior during washing and before diffu- 
sion occurs. Thus, on rearrangement of equation 7, the 
final internal cake liquor alumina weight must be 



A 4i = A oi e-tc + V ,(1 - e-tcj 



Aoi + A 3e 
V oi + V oe 



(8) 



This diffusion model is not the best one for giving good 
predictions. The diffusion model does not, for example, 
predict the washing efficiency increases (that is, R 
decreases) observed in the minipiant on passing from one 
wash stage to a succeeding one. Such increases are con- 
trary to what is typically reported in the literature, where 
washing efficiency decreases are considered common. 5 



'Dahlstrom, D. A. Predicting Performance of Continuous Filters. Chem. 
Eng. Prog., v. 74, No. 4, April 1978, pp. 69-74. 



< 
cr 



LlI 

o 

o 
o 



Cf 



'eq 



Initial concentration 




TIME (t) 
FIGURE 4.— Cake liquor solute concentration, C f , as a function of diffusion time. 



Furthermore, the use of equilibrium equations is rendered 
inconsequential on discovering that application of the 
model to miniplant data always requires that the positive 
time constant approach infinity. 



Shrinking Voids Model: Hypothetical Voids Shrinkage 

This model introduces the concept of an unwashable 
voids fraction that systematically changes (decreases) as 
the cake passes from one wash to the next. The model 
thus proposes a voids volume decrease that is directly pro- 
portional to the AI2O3 concentration decrease, AC (concen- 



tration decreases being a crude measure of viscosity 
decreases). Specifically, the new voids volume, V 4i , is ex- 
pressed in terms of the old voids volume, V oi , and a voids 
shrinkage constant, k 

V 4i =V oi -kAC. (9) 

Using the notation of figure 3, AC can be expressed as 

AA \ ( A oe + Aoi) - (A3e + A 3i) 



AC = 



V oi + v c 



(10) 



Since A oi = A 3i , this gives 
Aoe - A 3e 



AC = 



V i + V oe 

/Aoe - A 3e 
giving V 4i = V oi -k - — — 

\ v oi " v oe 



(11) 
(12) 



Since alumina concentration is uniform (at equilibrium) 
throughout the cake, the new voids alumina weight must 
be the voids volume fraction, V 4i /(V oi + Voe), of tne tota ' 
cake alumina, A 3 j + A3 e , 



A 4 i = 



V 4 i (A 3 i + A^) 

Voi + Vo« 



(13) 



The quantities V^ and A^ are then determined from 
material and volume balance as in figure 3. 



Completion of Countercurrent Calculations 

The essential relationships, summarized in figure 3, are 
applied to countercurrent washing beginning with a form 
cake of known composition, an estimated filtrate alumina 
weight, A 2 , and a filtrate volume equal to the wash water 
volume used in the system (refer to fig. 1). These calcula- 
tions are repeated for each successive wash and filtration 
stage. Then, at this point, it is generally found that the 
alumina weight, A f , in the final wash liquor does not equal 
the wash water weight originally specified (Af is usually 
zero). Therefore, the first filtrate alumina concentration, 
A 2 , must be reestimated and the whole countercurrent 
system balance calculation done again. After this second 
set of calculations is complete, the correct A 2 value may 
be closely estimated by linear interpolation 



A 2 = A 2 



„ (A f " • A f ) (A 2 " -A 2 Q 



(A f " - Af) 



(14) 



where the singly primed constants refer to wash water and 
first wash filtrate aluminas from the first material balance 
trial and the doubly primed numbers to those from the 
second trial. This interpolation always gives the correct A 2 
value for the diffusion model and generally comes close 
for the shrinking voids model. In any case, the TI-59 6 
calculator and Fortran programs used in doing these 
calculations do not assume that the equation yields exact 
answers, but rather use it to produce rapid convergence to 
the correct A 2 value. 

The foregoing calculations have presumed that the 
parameters, Vj, j, and tc (diffusion model) or V| and k 
(shrinking voids model) are known. If these are not known, 
the values must be varied in the search for another 
material balance that will cause the balance values to 
more closely approximate the correct values (measured 
during a filtration run). The closeness of fit will be opti- 
mized using a least squares method to be described at the 
end of the "Sample Calculations" section. 



Equations for Liquor Weight and Volume Balance 

Once the optimal set of parameters is established, 
liquor volumes and weight balances can be made. For- 
tunately, a simple method exists for converting alumina 
weight and liquor volume to liquor weight. For pure 
aqueous AICI 3 liquor, density is reliably calculated from 



lb \ / A lb Al 2 3 \ 

p — = 8.34 + 2.10 

gal / \V gal liquor/ 



(15) 



With impurities present, the constant 2.10 may be in- 
creased. 

With density known, the liquor weight then must be 

(W lb) = (V gal liquor) L — 
V gal 
= 8.34 (V gal liquor + 2.10 (A lb Al 2 3 ). (16) 



SUMMARY OF MATERIAL BALANCE PROCEDURE 

To obtain the optimum material balance for the two- 
wash belt filter shown in figure 1, it is necessary to 
measure the concentrations and volumes of specified 
filtration streams. This information is presented in table 1 
for miniplant test 1-3. The code 1 in the table specifies that 
the first wash filtrate is used to dilute the form filtration 
feed slurry. A code would indicate filtrate was not re- 
cycled. For the 123 pounds of dry leach solid passing 
through the filter each hour, the cake liquor volume can be 
analytically determined using the methods of appendix D. 
The parameters V, and k are merely trial values to use in 
the first set of shrinking voids model calculations. 

TABLE 1.— Miniplant belt filtration data, test 1-3 



Number of washes 

AUOa, weight-percent: 

1st wash filtrate 

Form cake liquor 

2d wash filtrate 

1st wash cake liquor 

Wash water 

2d wash cake liquor 

Code 

Reactor discharge: 

Al 2^3 pounds 

Liquor 1 gallons 

Wash water do 

Cake liquor, Vj do 

Internal cake liquor, Vj do 



square gallons per pound 



2.78 
8.31 
1.29 
6.84 


4.68 
1 

80.75 
74.40 
28.52 
12.76 
eg- 

e 5 



•Reference to specific equipment does not imply endorsement by the 
Bureau of Mines. 



Voids shrinkage constant, k 

e Estimated. 

'Includes 2 gal of flocculant water. 



The balance procedure assumes conservation of vol- 
ume as well as material. Therefore, all stream volumes (fig. 
5) are readily specified from the values in table 1, from the 
assumption that all wash filtrate volumes are the same as 
those for wash water, and from the assumption that all 
cake volumes, V t = V| + V e , are the same. 

The calculation begins with the determination of the 
form slurry alumina weight: 

Aform slurry = A reaC f 0r discharge + (Aist wash filtrate) (Code). 

The alumina weight for the two form filtration liquor 
fractions is proportional to volume, 

Vf orm fjitrate 

"form filtrate = tj x Af rm slurry anc ' 

v form slurry 

Aform cake = Af orm slurry — Af orm filtrate- 

Form cake alumina fractions (internal and external) are 
then 

Aj form cake — rr Af orm ca |< e and 
v t 

Ae form cake = Af orm ca |< e — Aj f orm C ake- 











(1) Reactor dischorge 


(5)First woeh filtrate 


7) Second wash filtrate 


(9) Wash water 




A = 


As 


As 


As 




W = 


W: 


W = 


Ws 






V: 


v= 


V= 


Vs 






^^^^^ Optional ^\. 






■ 




2) Flocculent 


(3) Form slurry 


\j 


A= 


A = 




W= 


W = 


/ist A ( 2nd "\ 
I wash ) \wQ9k J 




V = 


V = 










\ 

Form ^* 
_^. — — filtr. ^^ 






^ — — • ' \^^ 


<yS ' 


' 






(4) Form' filtrate 


(6) Form solids 


(8) First wash solids 


(10) Final wash solids 




A = 


As 


As 


As 




W = 


W = 


W= 


Ws 






Vs 


Vj = 
Ve = 


Vj = 

Ve = 


VjS 

Ve s 










Vt = 


V t s 


Vt « 




A= lb*. Al 2 3 W- lbs. liquor V= gal. liquor 


Vjsgal. internal coke liquor V e =gol external coke liquor V| = go 1. total coke liquor 


Conservation of A.Wond V observed 



FIGURE 5.— Sample filtration circuit material balance form. 



At this point, the form cake goes to the first wash and 
the figure 3 calculations are performed. First, N and f (for j 
= 1) must be calculated 



N = 



v wash liquor 
'external cake liquor 



and 



f = 1 - e-N. 



Then the wash liquor alumina is 

N 
Ai = rn( A 2" f Aoe). 
N-f 

Now the first wash solids alumina weight must be (by 
material balance) 

A 3 = Aq + A-, - A 2 . 



At this point, the cake undergoes voids shrinkage to 
give a new Vj 

V 3 i = V oi - 7 (Ao - A 3 ). 

Now if the current wash liquor, A-|, is not the last wash 
(wash water), the foregoing calculations are repeated 
beginning with internal-external alumina apportionment 
for the form cake. If, however, this is the last wash, A-| is 
compared to the known alumina content, Af. If the two are 
not identical (within 0.0005 lb), a new A 1st wash filtrate, A2, 
value must be applied for another series of calculations. 

Generally, after two sets of calculations, the material 
balance will not be obtained. Therefore, the following 
equation is used to determine the next A 2 estimate: 



A 2 = A 2 " - 



(V-AfW-A;/) 

(A f " - A f 



where the singly primed variables are values from the first 
calculation set and doubly primed variables are from the 
second set. 



SAMPLE CALCULATIONS 

First Material Balance Trial 

Using the data from table 1, along with the assumption 
that Ai s t wa sh filtrate = 16-00 lb, the form slurry alumina 
must be 

Aform slurry = 80.75 + (16.00)(1) = 96.75 lb. 
The two liquor fractions after form filtration are then 
90.16 



Aform filtrate — 



(96.75) = 84.75 lb and 



72.4 + 2.0 + 28.52 
Aform cake = 96.75 - 84.75 = 12.00 lb. 
The form cake liquor fractions are then 

9.00 

A| form cake = X 12.00 = 8.46 lb and 

12.76 

Ae form cake = 12.76 - 8.46 = 3.54 lb. 

For the next step, the wash ratio, N, and salt-free wash 
liquor alumina removal fraction, f, are 

28.52 

N = = 7.585 

3.76 

f - 1 _ e-7-585 _ 0.99949. 
The second wash filtrate alumina is then 

7.585(16.00 - 0.99949x3.54) 

A1 = A 2 a wash filtrate = - 



7.585 - 0.99949 
= 14.35 lb. 
Then the first wash cake solids alumina is 

Awashedcake = 12.00 + 14.35-16.00 = 10.35 1b. 



After voids shrinkage, the new Vj is 

9.00 - 5(12.00 - 10.35) 



'i washed cake — 



12.76 



= 8.357 gal. 



Now there is one more wash; therefore, the new cake 
aluminas are 

8.357 

An st wash cake = X 10.35 = 6.78 lb and 

12.76 

A e 1st wash cake = 10.35 — 6.78 = 3.57 lb. 

With a new external cake liquor volume, V e = 12.76 - 
8.357 = 4.403, the new N and f values must be 

N = 28.52/4.403 = 6.4776 and 

f _ 1 _ e -6.4776 _ Q.99846. 

The new wash filtrate (wash water this time) is then 

Awash water = Af 

6.4776(14.35 - 0.99846x3.57) 



6.4776 - 0.99846 

= 12.76 lb. 

This is, of course, 12.76 lb too much for wash water, and 
thus the entire calculation set must be repeated for a dif- 
ferent value of the first wash filtrate alumina, A 2 . 



Procedure Following First Balance Trial 

The first wash filtrate is now arbitrarily chosen to be 
8.00 lb. This time one obtains a wash water alumina value 
of 2.80 lb, as can be seen from the summaries in table 2. 

Now with two iterations completed, equation 14 can be 
used to obtain a good estimate for the true value of the 
first wash filtrate alumina 



(2.80 - 0)(8.00 - 16.00) 

Aist wash filtrate = 800 - = 5.75 lb. 

2.80 - 12.76 

After this series of calculations, the Al 2 3 weight in the 
wash water is found to be -0.03 lb. Further use of equation 
14 yields Ai st waS h filtrate as 5.78, which is the correct value 
for making the wash water alumina equal to 0.00 lb. 



Evaluation by Least Squares Error 

Goodness of Fit 

Now a least squares error evaluation of this balance 
can be done. The method used here is the sum of the 
squares of the fractional errors for the streams 



Z(error)2 = z 
i - 1 



i material balance 



- Ai 



analytical' 



.(17) 



analytical 



where the analytical weight can be reliably calculated 
from 



10 



TABLE 2.— Summary of a material balance calculation method, test 1-3 (Vj gal, k = 5 gal'/lb) 



Stream 



AI2O3, Pounds 



1st 
trial 



2d 
trial 



3rd 
trial 



Final 
trial 



Analysis 



Square of 
the error 



1st wash filtrate. A 2 

Feed slurry 

Form cake: 

Internal liquor 

External liquor 

2d wash filtrate 

1st wash cake: 

Internal liquor 

External liquor 

Wash water, A ( 

Final cake liquor 

Sum of the squares of the error 

ND Not determined. 
NAP Not applicable. 



16.00 
96.75 

8.46 

3.54 

14.35 

6.78 
3.57 

12.76 
ND 

NAP 



8.00 
88.75 

7.76 
3.24 
5.48 

5.33 
3.16 
2.80 
ND 
NAP 



5.75 
86.50 

7.57 
3.16 
2.98 

4.94 
3.02 
-.03 
ND 
NAP 



5.78 
86.53 

7.57 
3.16 
3.01 

4.94 
3.02 

4.95 
NAP 



7.04 
NAP 

10.73 
3.15 

8.53 


5.55 
NAP 



0.0320 
NAP 

.0000 
.0020 

.0045 

NAP 
.0117 
.0502 



^analytical 



= V x p x 



pet Al 2 3 
100 



(18) 

Wai 2 o 3 \ 
, 100 / 



= Vx8.34[1 + 0.02079 (pet AI 2 3 )1 1] 

This gives for the first wash filtrate 

/2.78\ 

Aanaiyticai = 28.52 x 8.34 [1 + 0.02079(2.78)1-1] _ 

W 
= 7.04 lb. 

Therefore, the square of the error is 

/5J8 - 7.04\2 
(error)2 = = 0.0320. 

\ 704 / 

The other errors are similarly calculated and presented in 
table 2. The sum of the squares of the error (SSE) for this 
material is thus 0.0502. But this SSE is, of course, not a 
least squares error, but rather just the error introduced on 
assuming specific values of Vj and k. 

Least Squares Fit 

To do a least squares fit, the variables V, and k must be 
changed until the SSE can no longer be lowered. This is a 



difficult task to handle; therefore, a TI-59 program was 
developed to reduce calculation time (see appendix B). 

Table 3 summarizes the SSE's obtained for various 
values of Vj and k. The best least squares fit was for Vj = 
9.2 gal and k = 7.5 ga|2/lb. 

If the shrinkage constant k is set to 0.0, the model is 
identical to the diffusion model with the number of cells j 
equal to 1. The least squares SSE is then 0.0902 for Vj = 
8.1 gal and the diffusion constant approaches infinity. 

When both the shrinkage constant and internal cake 
volume are 0.0, the model is identical to the PMCS model. 
The large SSE (2.1011) here compared with that for the 
shrinking voids model makes it obvious that the PMCS 
model is unsuitable for test 1-3 (and also for other tests). 



MODEL EVALUATION USING MINIPLANT DATA 

The shrinking voids model was applied to data from 
seven different miniplant tests where AICI 3 leach reactor 
discharges were filtered. The tests utilized three feed 
sizes— minus 10 mesh, minus 20 mesh, and minus 18 
mesh (misted). 

Misting is the process by which dust in the feed is sup- 
pressed. The as-mined raw kaolin fines are crushed to at 
least minus 14-mesh size and are then tumbled on an in- 
clined rotating disk, while being moistened by a fine mist 
of water. The momentary wetting of the outer surfaces of 
the large particles causes the submicroscopic particles 
present in the raw crushed kaolin to adhere to larger part- 
icles. Repeated impact of the particles against each other 



TABLE 3.— Summary of sums of squares of errors (SSE's) for test 1-3 material balance 



Vi, gal 


k, gal 2 /lb 







3.0 


5.0 


6.5 


7.0 


7.5 


8.0 


8.5 


10.0 


11.0 


0.9294 


0.7832 


0.6944 


0.6323 


0.6126 


0.5930 


0.5740 


0.5555 


0.5022 


10.5 


.6346 


.4888 


.4048 


.3486 


.3311 


.3143 


.2981 


.2826 


.2397 


10.0 


.4124 


.2798 


.2093 


.1658 


.1531 


.1413 


.1303 


.1202 


.0950 


9.5 


.2544 


.1448 


.0951 


.0697 


.0635 


.0584 


.0544 


.0514 


.0490 


9.3 


.2075 


.1093 


.0694 


.0524 


.0491 


.0471 


.0462 


.0464 


.0542 


9.2 


.1873 


.0953 


.0604 


.0478 


.0461 


'.04558 


.0463 


.0483 


.0612 


9.1 


.1692 


.0836 


.0540 


.0458 


.04563 


.0467 


.0491 


.0528 


.0709 


9.0 


.1531 


.0740 


2 .0499 


.0462 


.0471 


.0504 


.0545 


.0598 


.0833 


8.5 


.1010 


.0576 


.0619 


.0818 


.0915 


.1027 


.1153 


.1293 


.1798 


8.0 


3 .0913 


.0859 


.1197 


.1634 


.1813 


.2008 


.2219 


.2445 


.3213 


7.0 


.1722 


.2408 


.3296 


.4166 


.4493 


.4835 


.5196 


.5573 


.6796 


6.0 


.3473 


.4749 


.6050 


.7235 


.7666 


.8114 


.8580 


.9061 


1.0595 


5.0 


.5777 


.7398 


.8922 


1.0266 


1.0748 


1.1248 


1.1763 


1.2294 


1.3970 


3.0 


1.1122 


1.2609 


1.3970 


1.5159 


1.5585 


1.6024 


1.6477 


1.6942 


1.8411 





'2.1011 


2.0990 


2.1128 


2.1306 


2.1379 


2.1459 


2.1545 


2.1638 


2.1952 



'The least squares SSE, at V, = 9.2 and k = 7.5. 

'The SSE for the sample calculation differs from that in table 2 because of roundoff errors. 
The least squares SSE (0.0902) for the diffusion model is at Vj = 8.1, k = 0, and j = 1. 
The SSE for the PMCS model. 



11 



TABLE 4.— Material balances for seven miniplant tests 



Test and feed size 



1-3, 

minus 

10 mesh 



1-4, 

minus 

10 mesh 



3-2a, 

minus 

20 mesh 



3-3a, 

minus 

18 mesh' 



3-2b, 

minus 

20 mesh 



3-3b, 

minus 

18 mesh' 



3-4, 

minus 

10 mesh 



INPUT DATA 



Number of washes 

AUOj, weight percent: 

1st wash filtrate 

Form cake 

2d wash filtrate 

1st wash cake 

Wash water 

2d wash cake 

Code 

Reactor discharge: 

Al 2 3 pounds 

Liquor gallons 

Wash water do 

Cake liquor, gallons: 

Total 

Internal 

Voids shrinkage constant, k — square gallons per pound 



2 


2 


2 


2.78 


1.71 


2.80 


8.31 


7.37 


7.51 


1.29 


.77 


1.07 


6.84 


6.41 


5.64 











4.68 


3.78 


2.91 


1 


1 


1 


80.75 


78.76 


76.05 


74.40 


74.35 


73.96 


28.52 


42.64 


29.21 


12.76 


13.85 


12.93 


9.2 


10.0 


7.0 


7.5 


8.5 


-0.5 



2.42 
7.75 
1.06 
5.24 


3.35 

1 

76.65 
74.67 
28.52 

10.86 
6.5 
2.0 




3.55 
8.38 
2.81 
8.16 
2.30 
5.41 
1 

76.90 
74.40 
29.55 

13.20 
10.4 
10.5 



SHRINKING VOIDS MODEL MATERIAL BALANCE 



AUO3, pounds: 
1st wash filtrate 


5.822 
10.733 

3.231 

8.143 


4.911 
0.0456 


5.408 
9.964 
2.900 
7.457 

4.556 
0.4177 


6.937 
10.401 

2.758 

6.221 


3.463 
0.2100 


5.606 
8.657 
2.520 
5.571 

3.051 
0.0135 


9.341 
10.010 
5.998 
6.666 
3.024 
3.693 
0.0092 


7.973 
8.731 
4.772 
5.529 
2.440 
3.197 
0.0009 


9.601 


Form cake 


10.984 


2d wash filtrate 


8.032 


1st wash cake 


9 415 


Wash water 


5.963 


2d wash cake 


7 346 


Sum of squares of errors' 


0.0341 







'Misted. 

'Calculated using equation 17. 



as they are tumbled on the disk causes the small, dust-like 
particles to be smeared, or plastically molded 
onto the larger particles, so that the proportion of fines in 
the misted raw kaolin is greatly reduced, compared with 
raw crushed kaolin. Suppression of dust in the feed allows 
the leached slurry to both settle and filter rapidly. Less 
flocculant is needed, compared with the same size raw 
kaolin feed that has not been misted. 

The summaries in table 4 show the least squares error 
sums varying from 0.009 to 0.0456. If these sums are div- 
ided by the number of data streams used in calculating 
them (six streams), and the square root is then taken, the 
average stream error is seen to range from 1 .7 to 8.7 pet. 



There may be room for improvement, but this error value is 
much better than that for the PMCS model (64.8 pet) and 
the diffusion model (13.4 pet) in test 1-3. 

Analytical weights ana oest-tit balance weights are 
summarized in table 5. Comparison of best-fit (prior 
material balance) methods and shrinking voids model 
balances show the average model error (5.7 pet) to be 
somewhat superior to that for the best-fit average error (7.6 
pet). Thus, the shrinking voids model should be extremely 
useful for predicting material balances and predicting 
filtration performances in other plants. 

Probably the easiest approach to doing the material 
balance is through the Fortran program in appendix A. The 



TABLE 5.— Comparison of best-fit material balance with shrinking voids model material balance 



Test and feed size 



1-3, 

minus 

10 mesh 



1-4, 

minus 

10 mesh 



3-2a, 

minus 

20 mesh 



3-3a, 

minus 

18 mesh 



3-2b, 

minus 

20 mesh 



3-3b, 

minus 

18 mesh 



minus 
10 mesh 



ANALYTICAL WEIGHTS— UNBALANCED 



AloOg, pounds: 
1st wash filtrate 


7.04 
10.73 
3.15 
8.53 

5.55 


6.31 
10.11 
2.78 
8.59 

4.76 


7.26 
9.65 
2.67 
6.93 

3.35 


6.07 
8.41 
2.58 
5.36 

3.27 


9.77 
10.00 
5.70 
7.15 
3.02 
3.67 


7.93 
8.62 
4.69 
5.60 
2.44 
3.15 


9.48 


Form cake 


11.21 


2d wash filtrate 


7.37 


1st wash cake 


10.86 


Wash water 


5.96 


2d wash cake 


6.75 



BEST-FIT MATERIAL BALANCE 



AUO3, pounds: 
1st wash filtrate 


5.93 
10.83 
2.94 
7.84 

4.90 


5.81 
10.13 
3.31 
7.63 

4.32 


7.07 
10.05 
3.29 
6.27 

2.98 


6.02 
8.98 
2.39 
5.35 

2.96 


9.41 
10.26 
5.80 
6.65 
2.82 
3.67 


8.39 
9.05 
5.06 
5.72 
2.53 
3.19 


9.61 


Form cake 

2d wash filtrate 


11.11 
8.47 


1st wash cake 


9.97 


Wash water 

2nd wash cake 


5.95 
7.45 







AVERAGE PERCENT ERROR 2 



Best-fit 

Diffusion 

Shrinking voids 



8.5 

12.3 

8.7 



10.3 

12.0 

8.3 



11.4 
5.9 
5.9 



5.6 
6.5 
4.7 



4.4 
6.2 
3.9 



4.8 
2.2 
1.2 



8.2 
8.1 
7.5 



'Misted. 

'The overall average percent error is 7.6 for best-fit, 7.6 for diffusion, and 5.7 for the shrinking voids model. 



12 



following information must be obtained and organized into 
a data file, described in appendix A (program lines 
1200-1300). before running the program: 



Number of wash stages. 
Reactor discharge slurry. 

a. Weight-percent AI2O3. 

b. Volume (gallons). 
Flocculant water volume (gallons). 
Wash water volume (gallons). 
Weight-percent AI2O3— plant data. 

a. All wash filtrateliquors. 

b. All washed cake liquors. 

Cake liquor volume V t (gallons)— plant data. 
Optional flow code (0 = first wash filtrate not re- 
cycled, 1 = recycled). 



A sample printout for test 1-3 is given in appendix A ("Run 
in Material Balance Mode" section). 

In order to make the model into a useful predictor, it is 
now necessary to decide what factors determine the size 
of internal (Vj and total (Vt)) cake volumes. The most ob- 
vious factor is the nature of the solids. The three types of 
solids fed to the miniplant are characterized by the 
average cake liquor volumes and shrinkage constants in 
table 6. 



TABLE 6.— Average filtration parameters for miniplant 
leach residues containing 123 lb of dry solids 1 



Feed size, mesh 


V t ,gal 


Vi,gal 


k, gal 2 /lb 


Minus 10 

Minus 20 

Minus 18* 


13.27 
12.62 
10.60 


9.9 
6.9 
6.0 


8.8 
2.3 
1.8 



'All constants in the table are directly proportional to the weight of dry 
solids. For example, 246 lb of dry solids would have the following minus 
10-mesh feed parameters: V, = 26.54, Vj = 19.8, and k = 17.6. 

2 Misted. 



It should be noted that porosity measurements (appen- 
dix C) show that 123 lb of miniplant cake solids have a 
physical voids volume of 



known for three different sizes of leached kaolin slurry 
feeds. If it is now desired to predict belt filtration perfor- 
mance for another configuration using a different amount 
of wash water, the following information must be 
specified: 

1. Number of wash stages. 

2. Reactor discharge slurry. 

a. Weight-percent AI2O3. 

b. Volume (gallons). 

3. Flocculant water volume (gallons). 

4. Wash water volume (gallons). 

5. Feed cake parameters for a specified dry solid 
weight. 

a. V t gallons— lab or plant data. 

b. Vj gallons— modeling data. 

c. k square gallons per pound— modeling data. 

6. Optional flow code (0 = first wash filtrate not re- 
cycled, 1 = recycled). 

This information is then organized into the data file 
mentioned in appendix A, prior to running the Fortran pro- 
gram to give a complete material balance. 

Tables 7 and 8 summarize shrinking voids model 
predicted alumina losses and form filtrate (product) 
alumina concentrations and volumes obtainable using 
various operating configurations. The model may be also 
applied to filtration in plants processing other types of 
slurries. However, the appropriate salt-concentration-to- 
density conversions in the Fortran program must first be 
changed (see comments in program, appendix A) because 
the current program applies only for AICI 3 solutions. 



Optimization of Horizontal Belt Filtration Operations 

It is a general rule that countercurrent solids-liquid 
separation systems produce greater solute recoveries 
when more wash water is used. However, the extra water 
must often be removed later at additional expense. 
Recoveries may be also increased by adding extra wash 
stages— but at increased capital and operating costs. In 
any case, the final filtration configuration should be 



1 gal solids / 0.546 
V| = x 123 lb solids ( 



2.18x8.34 lb solids 
= 8.1 gal. 



,1-0.546 



Vj was also shown to be invariant with change in particle 
size. Therefore, these Vj's should be considered as effec- 
tive internal volumes rather than physical ones. In any 
case, these empirical values for each feed may be used to 
help predict filter performance. 



APPLICATION OF THE SHRINKING VOIDS MODEL 

Development of Material Balances From Plant Data 

Material balance calculation methods are detailed in 
the previous section. Simple application of the Fortran 
program used therein provides a balance more reliable 
than those of existing methods (best-fit, PMCS, and diffu- 
sion). The only restriction on the program is that new 
concentration-density conversions be applied when the 
solutions are not aqueous AICI3. 

Prediction of Hypothetical Plant Material Balances 

With the completion of a material balance derived from 
miniplant data, the parameters V t , Vj, and K become 



TABLE 7.— Predicted Al 2 3 filtration losses under 
various operating configurations, pounds 




Feed size, mesh 


washes 


Minus 10 Minus 20 Minus 18' 



20 GALLONS OF WASH WATER 



1 


8.256 


6.035 


5.114 


2 


5.787 


3.697 


3.084 


3 


3.834 


2.344 


1.890 


4 


2.481 


1.523 


1.172 


5 


1.582 


1.005 


.732 


6 


.991 


.669 


.458 





30 GALLONS OF WASH WATER 




1 


7.450 


5.335 


4.574 


2 


5.080 


3.013 


2.617 


3 


3.069 


1.685 


1.471 


4 


1.657 


.937 


.818 


5 


.802 


.518 


.451 


6 


.351 


.285 


.247 



50 GALLONS OF WASH WATER 



1 


6.240 


4.343 


3.820 


2 


4.238 


2.401 


2.129 


3 


2.506 


1.255 


1.148 


4 


1.285 


.641 


.606 


5 


.578 


.323 


.316 


6 


.235 


.162 


.163 



'Misted. 

NOTE.— Based on standard slurry feed: 77.62 lb of AI 2 Og, 76.52 gal liquor 
(which contains 5 gal of flocculant water). See appendix B, step 9, for 
method used in calculating these numbers on the TI-59. 



TABLE 8.— Predicted form filtrate concentrations 
under various operating configurations, 
weight-percent Al 2 3 1 



Number of 
washes 



Feed size, mesh 



Minus 10 



Minus 20 



Minus 18 2 



20 GALLONS OF WASH WATER 



1 


8.18 


8.36 


8.27 


2 


8.44 


8.61 


8.48 


3 


8.65 


8.75 


8.61 


4 


8.80 


8.84 


8.68 


5 


8.89 


8.89 


8.73 


6 


8.95 


8.93 


8.76 



30 GALLONS OF WASH WATER 



1 


7.46 


7.61 


7.54 


2 


7.68 


7.83 


7.72 


3 


7.88 


7.96 


7.83 


4 


8.01 


8.03 


7.89 


5 


8.10 


8.07 


7.92 


6 


8.14 


8.09 


7.94 





50 GALLONS OF WASH WATER 




1 


6.35 


6.46 


6.40 


2 


6.51 


6.62 


6.54 


3 


6.65 


6.71 


6.61 


4 


6.75 


6.76 


6.66 


5 


6.80 


6.79 


6.68 


6 


6.83 


6.80 


6.69 



'Calculated using weight-percent = 9.655 [(pounds of reactor discharge 
Al 2 3 — pounds of AUO3 loss)/(gallons form filtrate)] 1 ' 1 - 1 . 
2 Misted. 

NOTE.— Form filtrate volume was as follows: 





Feed size, mesh 


Wash water, 
gallons 


Minus 10 


Minus 20 


Minus 18 
Misted 


20 
30 

50 


83.25 

93.25 

113.25 


83.90 

93.90 

113.90 


85.92 

95.92 

115.92 



13 



chosen only after a series of material balances (like those 
summarized in tables 7 and 8) have been completed and 
the overall minimum plant cost determined. 

A selection from the three mesh sizes of feeds tested 
(minus 10, minus 20, and minus 18 misted) will, of course, 
be decided through the foregoing economic evaluation. 
However, table 7 shows that minus 18-mesh feed con- 
sistently yields the lowest alumina loss and minus 10- 
mesh feed the greatest. Since feed preparations are 
probably similar for these feed sizes, this seems to single 
out minus 18-mesh feed size as optimum. 

However, table 8 shows minus 18-mesh feed giving the 
poorest quality form filtrate product as the product is more 
dilute than for other feed solids. This poor quality results 
because the feed's small cake liquor volume, V t , removes 
significantly less water through the final wash cake than 
do other feeds. 

Large voids shrinkage constants, k, do not significantly 
contribute to improved Al 2 3 recovery even though they 
represent rapid improvement in washing efficiency. Note, 
for example, that k is largest for minus 10-mesh feed, 
which has the poorest recovery. 



SUMMARY AND CONCLUSIONS 



In order to minimize expensive in-plant testing of a con- 
tinuous horizontal washing and vacuum countercurrent 
belt filtration system, a model that can reliably predict 
system performance must be developed. Application of 
the PMCS filtration model to the miniplant data gave com- 
pletely unreliable results. Therefore, a shrinking voids 
model was proposed in which liquor residing in the cake 
particle pores is considered not readily washed from the 
cake. This model predicted increases in washing effi- 
ciency as the cake liquor becomes more dilute (and less 
viscous). The typical error in predicting stream alumina 
balance sets varied from 1.7 to 8.7 pet, with an average 
error of 5.7 pet. This method of balance was more effective 
than that for the diffusion model and for typical 
metallurgical best-fit methods. 



The optimum empirical parameters determined by 
shrinking voids model balances— V t , total cake liquor 
volume, Vj, internal cake liquor volume, and k, voids 
shrinkage constant— were then used to predict belt filtra- 
tion recoveries and product liquor concentrations for 
systems having various numbers of wash stages and 
various amounts of wash water. It was found that the 
minus 18-mesh misted solids were the best feed if op- 
timum alumina recoveries are desired. However, it was 
unclear as to what the optimum wash water volume or 
number of wash stages should be since this will depend 
on the economics for other steps in the clay-HCI- 
miniplant. 



14 

APPENDIX A.— MATERIAL BALANCE BY FORTRAN COMPUTER 

The Material Balance Program 

#FILE (BCME)BELFIL ON MRC 

1000 C-##### PROGRAM FOR DOING A BEST-FIT MATERIAL BALANCE ON DATA TAKEN 

1010 C-##### FROM A CONTINUOUS COUNTER-CURRENT BELT FILTRATION RUN. 

1020 C-##### THE BALANCE, DONE USING THE SHRINKING VOIDS MODEL, YIELDS 2 

1030 C-##### MODEL PARAMETERS, VI AND K, USEFUL IN FUTURE PREDICTIONS OF 

1040 C-##### FILTER PERFORMANCE. 

1050 C-##### WITH THESE PARAMETERS KNOWN, THE ONLY KNOWLEDGE REQUIRED 

1060 C-##### TO PREDICT FILTRATION PERFORMANCE IN OTHER CONFIGURATIONS IS 

1070 C-##### THE TOTAL CAKE LIQUOR VOLUME (VT). 

1080 C-##### BELT FILTRATION STREAM NUMBERING SYSTEM IS: 

1090 C-##### 1 - REACTOR DISCHARGE SLURRY 

1100 C-##### 2 - FLOCCULANT 

1110 C-##### 3 - FORM FILTRATION FEED SLURRY LIQUOR 

1120 C-##### 4 - FORM FILTRATE 

1130 C-##### 5 - 1ST WASH FILTRATE, 

1140 C-##### 6 - FORM FILTRATION CAKE LIQUOR 

1150 C-##### 7 - 1ST WASH LIQUOR = 2ND WASH FILTRATE 

1160 C-##### 8 - 1ST WASHED CAKE LIQUOR 

1170 C-##### 9 - 2ND WASH LIQUOR = 3RD WASH FILTRATE OR MINIPLANT WASH WATER 

1180 C-#####10 - 2ND WASHED CAKE LIQUOR 

1190 C— #####11 TO IW ETC.... 

1200 C-##### THIS PROGRAM READS DATA FROM FILE 5 = "BOLDAT" WHERE DATA IS 

1210 C-##### STORED IN THE FOLLOWING FORMATS: 

1220 C-##### FOR MATERIAL BALANCE MODE 

1230 C-##### 200 0, NO. OF WASHES, PCT AL203 IN REACTOR DISCHARGE 

1240 C-##### 300 PCT AL203 IN 1ST WASH FILTRATE LIQUOR 

1250 C-##### 305 PCT AL203 IN 1ST WASHED CAKE LIQUOR 

1260 C-##### 310 PCT AL203 IN 2ND WASH FILTRATE LIQUOR 

1270 C-##### 315 PCT AL203 IN 2ND WASHED CAKE LIQUOR 

1280 C-##### 320 ETC. UNTIL ALL PLANT DATA HAS BEEN ENTERED 

1290 C-##### 400 GALLONS REACTOR DISCH., GAL. FLOCCULENT ADDED TO DISCH., 

1300 C-##### GAL. WASH WATER, TOTAL GAL. LIQUOR IN CAKE, 

1310 C-##### A CODE( = 1ST WASH FILTRATE NOT RECYCLED, 1=RECYCLED) 

1320 C-##### 

1330 C-##### FOR PREDICTOR MODE 

1340 C-##### 200 1, NO. OF WASHES, PCT AL203 IN REACTOR DISCHARGE 

1350 C-##### 300 GAL INTERNAL CAKE LIQUOR, VOIDS SHRINKAGE CONSTANT 

1360 C-##### 400 (SAME AS FOR MATERIAL BALANCE ABOVE) 

1370 C-##### 

1380 C-##### IF SOLUTIONS ARE NOT AQUEOUS ALCL3 THEN DENSITY EQUATIONS 

1390 C-##### MUST BE CHANGED AT POINTS MARKED BY " . 

1400 C-##### LIST OF VARIABLES USED IN PROGRAM: 

1410 C-##### ANCON(X)— MEASURED AL203 CONCENTRATION (PCT.) FOR STREAM X 

1420 C-##### ANWT(X)— MEASURED AL203 WEIGHT (LBS.) FOR STREAM X 

1430 C-##### AWT(X)— TRIAL VALUE AL203 WEIGHT (LBS) FOR STREAM X 

1440 C-##### WTLIQ(X)— WEIGHT (LBS) FOR LIQUOR IN STREAM X 

1450 C-##### VOLLIQ(X)— VOLUME (GALLONS) FOR LIQUOR STREAM X 

1460 C~##### AIWT(X)— INTERNAL LIQUOR AL203 WEIGHT (LBS), STREAM X 

1470 C-##### AEWT(X)— EXTERNAL LIQUOR AL203 WEIGHT (LBS), STREAM X 



15 



1480 C-##### NOWASH-- NUMBER OF WASH STAGES 

1490 C-##### VOLFD-- VOLUME OF FEED STREAM, NO. 1 

1500 C-##### VOLFL-- VOLUME OF FLOCCULANT ADDITION STREAM 

1510 C-##### VOLWW-- VOLUME OF WASH WATER STREAM 

1520 C-##### VOLCL-- CAKE LIQUOR VOLUME (INTERNAL + EXTERNAL) 

1530 C-##### CODE— (1 FOR RECYCLE OF 1ST WASH FILTRATE, FOR NO RECYCLE) 

1540 C-##### IW— TOTAL NO. OF STREAMS 

1550 C-##### L— SIZE OF VI - CONST TEST MATRIX 

1555 C-##### SMALL L's REDUCE CALCULATION TIME, BUT MAY DECREASE ACCURACY 

1560 C-##### VI— 1ST CAKE INTERNAL VOLUME, REFERENCE VALUE 

1570 C-##### VIC— ANY CAKE INTERNAL VOLUME, CALCULATED 

1580 C-##### DVI— INCREMENTAL CHANGE IN TRIAL VALUE OF VI 

1590 C-##### BVI— BEST VALUE FOR VI 

1600 C-##### EK— VOIDS SHRINKAGE CONSTANT, REFERENCE VALUE 

1610 C-##### EKK— VOIDS SHRINKAGE CONSTANT, CURRENT VALUE 

1620 C-##### DKK— INCREMENTAL CHANGE IN TRIAL VALUE OF EKK 

1630 C-##### BKK— BEST VALUE FOR EKK 

1631 C-##### WASHRA— RATIO OF WASH WATER VOLUME TO CAKE VOLUME 

1632 C-##### F— FRACTION OF AL203 REMOVED BY AL203-FREE WASH WATER 
1640 C-##### AWT5C— STREAM 5 AL203 WT., CURRENT VALUE 

1650 C-##### AWT5E— STREAM 5 AL203 WT., EARLIER VALUE 

1660 C-##### WWC— WASH WATER AL203 WT., CURRENT VALUE 

1670 C-##### WWE— WASH WATER AL203 WT., EARLIER VALUE 

1680 C-##### SSE— SUM OF SQUARES OF ERRORS 

1690 C-##### ASE— AVERAGE SUM OF SQUARE OF ERROR PER STREAM 

1700 C-##### BASE— BEST AVG. SUM OF SQUARE OF ERROR 

1710 FILE 5(KIND=DISK,TITLE="B0LDAT",FILETYPE=7) 

1720 DIMENSION ANCON(50) ,ANWT(50) ,AWT(50) ,WTLIQ(50) , 

1730 -VOLLIQC 50) , AIWT ( 50) , AEWT ( 50) 

1740 C-##### MODE DOES BEST-FIT MATERIAL BALANCE, MODE 1 PREDICTS A BALANCE 

1750 READ(5,8) MODE,NOWASH,ANCON(l) • 

1760 8 FORMAT ( 214, F8. 3) 

1770 WRITE(6,9) MODE,NOWASH,ANCON(l) 

1780 9 FORMAT(5H MODE,I3/I3,35H WASHES, REACTOR DISCH.AL203 CONC.=,F8.3) 

1790 IW=NOWASH*2+6 

1800 IF(MODE.EQ.l) GOTO1200 

1810 IF(MODE.NE.O) GOTO 2000 

1820 JW=IW 

1830 L=6 

1840 N=4 

1850 10 N=N+1 

1860 READ(5,11) ANCON(N) 

1870 11 FORMAT (F8. 3) 

1880 IF(N.LT.IW) GOTO 10 

1890 WRITE(6, 12) ( (I , ANCON(I) ) , 1=5, IW) 

1900 12 FORMAT(7H STREAM,I3,F8.3,11H PCT. AL203) 

1910 ANCON(4)=ANCON(6) 

1920 ANCON(3)=ANCON(6) 

1930 ANCON(2)=0 

1940 20 READ(5,21) VOLFD,VOLFL,VOLWW,VOLCL,CODE 

1950 21 FORMAT(5F8.3) 

1960 WRITE (6, 22) VOLFD,VOLFL,VOLWW,VOLCL,CODE 

1970 22 FORMAT (/19H VOLUMES IN GALLONS/ 65H REACTOR DISCHARGE FIjOCCULANT 



16 

1980 -WASH WATER CAKE LIQUOR CODE/5F13.2) 

1990 IF(CODE.NE.O.O.AND.OODE.NE.l.O) GOTO 2000 

2000 IF(IW.LT.8.0R.IW.GT.50) GOTO 2000 

2010 IT=0 

2020 VOLLIQC 1 ) =VOLFD 

2030 VOLLIQC 2 )=VOLFL 

2040 C-##### INITIALIZES VOLUMES FOR FILTRATES & CAKE LIQUORS 

2050 DO 50 1=6, IW, 2 

2060 VOLLIQC 1-1 )=VOLWW 

2070 VOLLIQC I ) =VOLCL 

2080 50 CONTINUE 

2090 VOLLIQC3)=VOLFD+VOLFL+VOLLIQ(5)*CODE 

2100 VOLLIQC 4 )=VOLLIQ( 3) -VOLLIQC 6) 

2130 C = CALCULATES AL203 WTS. USING AL203 CONCENTRATIONS 

2140 DO 60 1=1, JW 

2150 ANWTCI)=VOLLIQ(I)*.0834*C1+.02079*ANCONCI)**1.1)*ANCONCI) 

2160 60 CONTINUE 

2170 IFCMODE.EQ.l) GOTO 70 

2 180 DVI=VOLCL/ C L-l ) 

2190 VI=0 

2200 DKK=V0LFD**2/CCL-1)*ANWTC1)*N0WASH) 

2210 EK=-2*DKK/(L-1) 

2220 BASE=99999 

2230 70 AWT(1)=ANWTC1) 

2240 AWT(2)=0 

2250 C-##### BEGINS THE BEST-FIT MATERIAL BALANCE 

2260 80 DO 1100 1=1, L 

2270 EKK=EK 

2280 DO 1000 J=1,L 

2290 C-##### VI >= VOLCL IS UNREASONABLE, CAUSING CALCULATION PROBLEMS 

2300 IF( VI .GE . VOLCL) VI= .99* VOLCL 

2310 AWTC5)=8 

2320 AWT5C=9 

2330 WWC=ANWT C IW-1 ) +1 

2340 AWT5E=AWT5C+1 

2350 WWE=WWC+1 

2360 LIM=0 

2370 C-##### DOES 1 MATERIAL BALANCE FOR ASSUMED 1ST FILTRATE COMPOSITION 

2380 799 LIM=LIM+1 

2390 VIC=VI 

2400 AWTC3)=AWTC1)+AWT(2)+AWTC5)*C0DE 

2410 AWT(6)=VOLLIQC6)*AWTC3)/VOLLIQC3) 

2420 AIWTC6)=VI*AWTC6)/VOLLIQC6) 

2430 AEWTC6)=AWT(6)-AIWTC6) 

2440 DO 900 K=7,IW,2 

2450 WASHRA=VOLWW/(VOLCL-VIC) 

2460 F=1-EXP(-WASHRA) 

2470 AWTCK)=WASHRA*(AWTCK-2)-F*AEWTCK-l) )/CWASHRA-F) 

2480 TEMP=AEWTCK-l)+AWTCK)-AWT(K-2) 

2490 VIC=VIC-EKK* CAEWT(K-l)-TEMP)/VOLLIQCK-l) 

2500 C-##### VIC >= VOLCL IS UNREASONABLE, CAUSING CALCULATION PROBLEMS 

2510 IF( VIC .GE. VOLCL) GOTO 999 

2520 AIWT(K+l)=VIC*(AIWTCK-l)+TEMP)/VOLLIQCK-l) 



17 



2530 AEWT(K44)=AIWT(K-1)+TEMP-AIWT(K+1) 

2540 AWT(K+1)=AIWT(K+1)+AEWT(K+1) 

2550 900 CONTINUE 

2560 AWT5E=AWT5C 

2570 AWT5C=AWT(5) 

2580 WWE=WWC 

2590 WWC=AWT(IW-1) 

2600 ERR= ABS ( ANWT ( IW-1 ) -AWT ( IW-1 ) ) 

2610 IF( ERR. LT. 0.0005) GOTO 950 

2620 IF(LIM.GT.20) GOTO 999 

2630 C-##### WASH WATER AL203 DIFFERS FROM TRUE AL203 BY MORE THAN 0.0005 

2640 C-##### LBS. SO, CHOOSE NEW 1ST FILTRATE ESTIMATE 

2650 AWT ( 5 ) =AWT5C-( WWC-ANWT ( IW-1 ) ) * ( AWT5C-AWT5E ) / ( WWC-WWE ) 

2660 GOTO 799 

2670 950 AWT(4)=AWT(3)-AWT(6) 

2680 IT=IT+1 

2690 IF(MODE.EQ.l) GOTO 963 

2700 C-##### CALCULATES SUMS OF SQUARES OF ERRORS ON PCT. BASIS 

2710 SSE=0 

2720 NUM=0 

2730 DO 960 11=5, IW 

2740 IF(ANWT(II).EQ.0.0)GOTO 960 

2750 SSE=SSE+((ANWT(II)-AWT(II))/ANWT(II))**2 

2760 NUM=NUM+1 

2770 960 CONTINUE 

2780 ASE=SSE/NUM 

2790 IF(ASE.GT.BASE) GOTO 962 

2800 C-##### A LOWER SSE HAS BEEN FOUND 

2810 BASE=ASE 

2820 BVI=VI 

2830 BKK=EKK 

2840 962 IF(MODE.NE.l) GOTO 999 

2850 C-##### PRINT THE FINAL RESULTS 

2860 963 WRITE(6,964) 

2870 964 FORMAT(/15X,26HTHE FINAL MATERIAL BALANCE) 

2880 WRITE(6,965) VI,EKK 

2890 965 FORMAT(/27H PARTICLE INTERNAL VOLUME »,F8.3, 

2900 - 30H GALLONS, SHRINKAGE CONSTANT =,F8.3) 

2910 WRITE(6,970) 

2920 970 FORMAT (45H NO. LBS. AL203 LBS. LIQUOR GAL. LIQUOR) 

2930 DO 980 M=1,IW 

2940 C-=— == CALCULATES WEIGHT OF AQUEOUS ALCL3 SOLUTIONS 

2950 WTLIQ(M)=8.34*VOLLIQ(M)+2.10*AWT(M) 

2960 WRITE(6,985)M,AWT(M) ,WTLIQ(M) ,VOLLIQ(M) 

2970 980 CONTINUE 

2980 985 FORMAT(I4,F12.3,2F12.2) 

2990 WRITE(6,990)ASE,SSE 

3000 990 FORMAT(20H AVG. SQ. OF ERROR = ,F9.6,7H SSE =,F9.5) 

3010 WRITE(6,995) IT 

3020 995 FORMAT (19H NO. OF BALANCES = ,14) 

3030 999 EKK=EKK+DKK 

3040 1000 CONTINUE 

3050 VI=VI+DVI 



18 



3060 1100 CONTINUE 

3070 IF(MODE.EQ.l) GOTO 2000 # 

3080 Q-mmn CHOOSES area OF VI - CONST matrix for next search 

3090 DVI=DVI/2 

3100 VI=BVI-(L-l)*DVI/2 

3110 DKK=DKK/2 

3120 EK=BKK-(L-l)*DKK/2 

3130 IF(DVI.GT.O.Ol.OR.DKK.GT.O.l) GOTO 80 

3140 C-##### BEST-FIT HAS BEEN POUND, PREPARE TO PRINT IT 

3150 MODE=l 

3160 L=l 

3170 VI=BVI 

3180 EK=BKK 

3190 GOTO 80 

3195 C-##### PREDICTOR MODE INITIALIZATION 

3200 1200 READ(5,1700) VI, EK 

3210 1700 FORMAT(2F8.3) 

3220 WRITE(6,1800) VI, EK 

3230 1800 FORMAT(23H INTERNAL CAKE VOLUME =,F8.3,17H GALLONS, CONST.=,F8.3) 

3240 L=l 

3250 JW=1 

3260 AWT(IW-1)=0 

3270 GOTO 20 

3280 2000 STOP 

3290 END 

Run in Material Balance Mode (Test 1-3) 



LIST BOLDAT 

#FILE (BCME)BOLDAT ON MRC 

100 2 10.483 

200 2.78 

300 8.31 

400 1.29 

500 6.84 

600 

700 4.68 

800 72.4 2. 

# 

RUN BELFIL 

^RUNNING 0588 



28.52 12.76 1. 



MODE 
2 WASHES, 



STREAM 
STREAM 
STREAM 
STREAM 
STREAM 



5 
6 
7 
8 
9 



STREAM 10 



REACTOR DISCH.AL203 CONC.= 
2.780 PCT. AL203 
8.310 PCT. AL203 
1.290 PCT. AL203 
6.840 PCT. AL203 
0.000 PCT. AL203 
4.680 PCT. AL203 



10.483 



VOLUMES IN GALLONS 
REACTOR DISCHARGE 
72.40 



FLOCCULANT 
2.00 



WASH WATER 
28.52 



CAKE LIQUOR CODE 

12.76 1.00 



19 

The Final Material Balance 

PARTICLE INTERNAL VOLUME = 9.141 GALLONS, SHRINKAGE CONSTANT = 7.065 
NO. LBS. AL203 LBS. LIQUOR GAL. LIQUOR 



1 




80.747 


773.39 


72.40 


2 




0.000 


16.68 


2.00 


3 




86.595 


1040.20 


102.92 


4 




75.859 


911.24 


90.16 


5 




5.848 


250.14 


28.52 


6 




10.736 


128.96 


12.76 


7 




3.212 


244.60 


28.52 


8 




8.100 


123.43 


12.76 


9 




0.000 


237.86 


28.52 


10 




4.888 


116.68 


12.76 


AVG. 


SQ. 


OF ERROR 


= 0.009120 


SSE = 0.04560 


NO. 


OF BALANCES = 


289 




#ET=43.0 


PT=1.5 10=0.3 





Run in Predictor Mode (Four Washes) 

LIST BOLDAT 

#FILE (BCME) BOLDAT ON MRC 

100 1 4 10.254 

200 9.9 8.8 

300 71.52 5. 20. 13.27 1. 

# 

RUN BELFIL 

^RUNNING 0679 

MODE 1 

4 WASHES, REACTOR DISCH.AL203 CONC.= 10.254 
INTERNAL CAKE VOLUME = 9.900 GALLONS, CONST .= 8.800 

VOLUMES IN GALLONS 

REACTOR DISCHARGE FLCCCULANT WASH WATER CAKE LIQUOR CODE 

71.52 5.00 20.00 13.27 1.00 



20 

The Final Material Balance 

PARTICLE INTERNAL VOLUME = 9.900 GALLONS, SHRINKAGE CONSTANT = 8.800 
NO. LBS. AL203 LBS. LIQUOR GAL. LIQUOR 



1 




77.619 


759.48 


71.52 


2 




0.000 


41.70 


5.00 


3 




87.115 


987.92 


96.52 


4 




75.138 


852.09 


83.25 


5 




9.496 


186.74 


20.00 


6 




11.977 


135.82 


13.27 


7 




7.768 


183.11 


20.00 


8 




10.249 


132.19 


13.27 


9 




5.563 


178.48 


20.00 


10 




8.044 


127.56 


13.27 


11 




2.904 


172.90 


20.00 


12 




5.385 


121.98 


13.27 


13 




-0.000 


166.80 


20.00 


14 




2.481 


115.88 


13.27 


VG. 


SQ. 


OF ERROR = 


0.000000 


SSE = 0.00000 


0. 


OF BALANCES = 


1 





#ET=45.9 PT=0.5 IO=0.4 



21 



APPENDIX B.— MATERIAL BALANCE USING A 
PROGRAMMABLE CALCULATOR 



The balance form in figure 5 is used for this procedure. 

1. Input composition is specified (A, W, V) for 

a. Heat exchanger (reactor) discharge 

b. Flocculant 

c. Wash water 

2. All wash filtrate volumes are set equal to the wash 
water volume. 

3. An average cake liquor volume is determined from 
the average value obtained in the miniplant run. See 
appendix C for methods used to determine this 
value. Set all cake volumes equal to this value V t . 

4. Make estimates (guesses) for values of 

a. Internal cake liquor volume (Vj = to V t ) 

b. Voids shrinkage constant (k = any finite 
value) 

5. Make up the following data table specifying the 
values and order of data entry into the TI-59 program- 
mable calculator: 

a. Number of washes (1 to 12) n 

b. First wash filtrate, analytical 

wt-pct Al 2 3 a-i 

c. First feed cake (= form cake), 

analytical wt-pct Al 2 3 a 2 

d. First wash liquor, analytical 

Wt-pct AI2O3 83 

e. Succeeding feed cakes (if applicable) 

f. Succeeding wash liquors (if applicable) 

g. Final feed cake, analytical 

wt-pct Al 2 3 a 4 

h. Final wash liquor (that is, wash water), 

analytical wt-pct AI2O3 a 5 

i. Final product cake, analytical 

wt-pct AI2O3 a6 

j. Circuit code (0 = first wash filtrate 

not recycled, 1 = recycled) c 

k. Reactor discharge, pounds of Al 2 3 .... A 2 
I. Reactor discharge, gallons of liquor .... V 2 

m. Wash water, gallons of liquor V f 

n. Total cake liquor, average gallons 

of liquor V t 

o. Internal cake liquor, gallons 

(estimated) Vj 

p. Voids shrinkage constant, square 

gallons per pound k 

6. The TI-59 program summarized at the end of this ap- 
pendix is run as follows: 

a. Push RST button 

b. Push R/S, enter first data value from step 5 (data 
table). Repeat until entire table has been 
entered. 

c. After a few minutes, the analytical weight 
balances for streams of steps 5b through 5i will 
be printed out in order. The final SSE value 
represents the sum of the percent least squares 
errors for the previous analytical balance values. 

7. To find a least squares SSE, different values of Vj 
and k must be tried. To change these values, enter 
them in order after the last SSE is read and another 
material balance will be printed out. Continue to do 
this until SSE reaches a minimum value. 



To save printer paper, keep printer off until it has 
been determined that a minimum SSE has been 
reached. Then turn the printer back on and enter the 
appropriate Vj and k values to obtain the optimal 
Al 2 3 balance values. 

At this time, one may want to predict a balance for a 
filtration system having different values for different 
reactor discharge alumina (memory 31), reactor li- 
quor volumes (memory 30), wash water volumes 
(memory 29), cake liquor volumes (memory 28), wash 
water alumina weights (memory 2), and numbers of 
wash stages (memory 42). If so, enter these new 
values into the corresponding memories. If the first 
wash filtrates in memories 34 and 36 are equal, 
change the values of one of them. Then enter the 
number of washes into the register, push A, push A', 
and enter values for steps 5o and 5p into calculator 
and wait for the calculator to stop. Then push RCL 
40 to obtain the final cake alumina losses. 

Figure B-1 provides a schematic of the program 
for material balance using the shrinking voids 
model. 



Program Memory Bank 



1 A ca k e analysis (final) 

2 A wash ii quor analysis (wash water) 

3 A cake analysis (next to last) 

4 A wasn |jq U or analysis (next to last) 

5-24 Alternate A ca ke analysis ar| d A wasn |jq UOr analysis 

until first stage analysis is reached 

25 Not assigned 

26 Af j rS ( W ash liquor guess 

27 Vjnternal cake liquor guess 

28 Vjotal cake liquor 

29 V wa sh water 

30 V rea ctor discharge + flocculant 

31 A reac tor discharge + flocculant 

32 Circuit code 

33 A r ', wash water (earlier calculation) 

34 A2', first wash filtrate (earlier calculation) 

35 Af ", wash water (current calculation) 

36 A 2 ", first wash filtrate (current calculation) 

37 A t otai cake liquor (previous wash) 

38 A ex t e rnal feed cake 

39 Asternal feed cake 

40 A t otal cake liquor (present wash) 

41 k, voids shrinkage constant 

42 Number of washes 

43 N, wash ratio 

44 f, pure water wash recovery 

45 Vj n t e rnal cake liquor 

46 I(error)2 

47 Current wash calculation number 

48 Memory where current A stored 

49 Location of highest numbered A 



22 



Belt-Filter Material Balance Modeling 















TI-59 


Program 












000 


22 


INV 


050 


72 


ST* 


100 


36 


36 


150 


65 


X 


200 


43 RCL 


001 


58 


FIX 


051 


00 


00 


101 


42 


STO 


151 


43 


RCL 


201 


44 44 


002 


91 


R/S 


052 


69 OP 


102 


26 


26 


152 


32 


32 


202 


95 = 


003 


11 


A 


053 


30 


30 


103 


25 CLR 


153 


95 


= 


203 


42 STO 


004 


22 


INV 


054 


71 


SBR 


104 


42 


STO 


154 


42 


STO 


204 


35 35 


005 


86 


STF 


055 


45 


Y* 


105 


47 


47 


155 


40 


40 


205 


43 RCL 


006 


01 


01 


056 


43 


RCL 


106 


43 


RCL 


156 


65 


X 


206 


42 42 


007 


42 


STO 


057 


28 


28 


107 


45 


45 


157 


43 


RCL 


207 


32 X:T 


008 


00 


00 


058 


95 


= 


108 


42 


STO 


158 


27 


27 


208 


43 RCL 


009 


42 


STO 


059 


72 


ST* 


109 


27 


27 


159 


55 


4 


209 


47 47 


010 


36 


36 


060 


00 


00 


110 


94 


+/- 


160 


43 


RCL 


210 


67 EQ 


Oil 


85 


+ 


061 


97 


DSZ 


111 


85 


+ 


161 


28 


28 


211 


13 C 


012 


01 


1 


062 


00 


00 


112 


43 


RCL 


162 


95 


= 


212 


87 IFF 


013 


95 


= 


063 


61 


GTO 


113 


28 


28 


163 


42 


STO 


213 


00 00 


014 


42 


STO 


064 


76 


LBL 


114 


95 


= 


164 


39 


39 


214 


38 SIN 


015 


34 


34 


065 


16 


A» 


115 


35 


1/X 


165 


94 


+/- 


215 


43 RCL 


016 


76 


LBL 


066 


91 


R/S 


116 


65 


X 


166 


85 


+ 


216 


35 35 


017 


15 


E 


067 


42 


STO 


117 


43 


RCL 


167 


43 


RCL 


217 


19 D' 


018 


91 


R/S 


068 


45 


45 


118 


29 


29 


168 


40 


40 


218 


76 LBL 


019 


72 


ST* 


069 


04 


4 


119 


95 


= 


169 


95 


= 


219 


38 SIN 


020 


00 


00 


070 


02 


2 


120 


42 


STO 


170 


42 


STO 


220 


43 RCL 


021 


97 


DSZ 


071 


02 


2 


121 


43 


43 


171 


38 


38 


221 


35 35 


022 


00 


00 


072 


04 


4 


122 


94 


+/- 


172 


43 


RCL 


222 


48 EXC 


023 


15 


E 


073 


69 OP 


123 


22 


INV 


173 


40 


40 


223 


26 26 


024 


91 


R/S 


074 


04 


04 


124 


23 


LNX 


174 


87 


IFF 


224 


94 +/- 


025 


42 


STO 


075 


43 


RCL 


125 


94 


+/- 


175 


00 


00 


225 


85 + 


026 


32 


32 


076 


45 


45 


126 


85 


+ 


176 


14 


D 


226 


43 RCL 


027 


91 


R/S 


077 


69 


OP 


127 


01 


1 


177 


19 


D' 


227 


26 26 


028 


42 


STO 


078 


06 


06 


128 


95 


= 


178 


76 


LBL 


228 


85 + 


029 


31 


31 


079 


02 


2 


129 


42 STO 


179 


14 


D 


229 


43 RCL 


030 


91 


R/S 


080 


06 


6 


130 


44 


44 


180 


01 


1 


230 


38 38 


031 


42 


STO 


081 


69 OP 


131 


43 


RCL 


181 


44 


SUM 


231 


85 + 


032 


30 


30 


082 


04 


04 


132 


31 


31 


182 


47 


47 


232 


43 RCL 


033 


91 


R/S 


083 


91 


R/S 


133 


85 


+ 


183 


43 


RCL 


233 


39 39 


034 


42 


STO 


084 


42 STO 


134 


43 


RCL 


184 


26 


26 


234 


95 = 


035 


29 


29 


085 


41 


41 


135 


26 


26 


185 


75 


— 


235 


48 EXC 


036 


91 


R/S 


086 


69 OP 


136 


65 


X 


186 


43 


RCL 


236 


40 40 


037 


42 


STO 


087 


06 


06 


137 


43 


RCL 


187 


44 


44 


237 


42 STO 


038 


28 


28 


088 


86 


STF 


138 


32 


32 


188 


65 


X 


238 


37 37 


039 


43 


RCL 


089 


00 


00 


139 


95 


= 


189 


43 


RCL 


239 


75 - 


040 


49 


49 


090 


43 


RCL 


140 


65 


X 


190 


38 


38 


240 


43 RCL 


041 


42 


STO 


091 


49 


49 


141 


43 


RCL 


191 


95 


= 


241 


40 40 


042 


00 


00 


092 


42 


STO 


142 


28 


28 


192 


65 


X 


242 


95 = 


043 


76 


LBL 


093 


48 


48 


143 


55 


4 


193 


43 


RCL 


243 


65 x 


044 


61 


GTO 


094 


25 CLR 


144 


53 


( 


194 


43 


43 


244 


43 RCL 


045 


71 


SBR 


095 


42 


STO 


145 


43 


RCL 


195 


55 


T 


245 


41 41 


046 


45 


YX 


096 


46 


46 


146 


30 


30 


196 


53 


( 


246 


55 « 


047 


43 


RCL 


097 


76 


LBL 


147 


85 


+ 


197 


43 


RCL 


247 


43 RCL 


048 


29 


29 


098 


12 


B 


148 


43 


RCL 


198 


43 


43 


248 


28 28 


049 


95 


= 


099 


43 


RCL 


149 


29 


29 


199 


75 


— 


249 


95 = 



23 



250 


22 


INV 


304 


67 BQ 


358 


48 


EXC 


412 


46 


46 


466 65 x 


251 


44 SUM 


305 


14 D 


359 


36 


36 


413 


01 


1 


467 02 2 


252 


27 


27 


306 


03 3 


360 


48 


EXC 


414 


22 


INV 


468 85 + 


253 


43 


RCL 


307 


06 6 


361 


35 


35 


415 


44 


SUM 


469 02 2 


254 


29 


29 


308 


03 3 


362 


48 


EXC 


416 


48 


48 


470 95 = 


255 


55 


• 


309 


06 6 


363 


33 


33 


417 


76 


LBL 


471 42 STO 


256 


53 


( 


310 


01 1 


364 


48 


EXC 


418 


52 


EE 


472 49 49 


257 


43 


RCL 


311 


07 7 


365 


35 


35 


419 


92 


RTN 


473 92 RTN 


258 


28 


28 


312 


69 OP 


366 


43 


RCL 


420 


76 


LBL 




259 


75 


— 


313 


04 04 


367 


36 


36 


421 


17 


B' 




260 


43 


RCL 


314 


43 RCL 


368 


75 


- 


422 


43 


RCL 




261 


27 


27 


315 


46 46 


369 


53 


( 


423 


35 


35 




262 


95 


= 


316 


69 OP 


370 


43 


RCL 


424 


99 


PRT 




263 


42 STO 


317 


06 06 


371 


35 


35 


425 


01 


1 




264 


43 


43 


318 


61 GTO 


372 


75 


— 


426 


22 


INV 




265 


94 


+/- 


319 


16 A' 


373 


43 


RCL 


427 


44 


SUM 




266 


22 


INV 


320 


76 LBL 


374 


02 


02 


428 


48 


48 




267 


23 


LNX 


321 


13 C 


375 


54 


) 


429 


61 


GTO 




268 


94 


+/- 


322 


22 INV 


376 


55 


4 


430 


38 


SIN 




269 


85 


+ 


323 


87 IFF 


377 


53 


( 


431 


76 


LBL 




270 


01 


1 


324 


00 00 


378 


43 


RCL 


432 


45 


Y x 




271 


95 


= 


325 


17 B' 


379 


35 


35 


433 


01 


1 




272 


42 


STO 


326 


58 FIX 


380 


75 


— 


434 


85 


+ 




273 


44 


44 


327 


03 03 


381 


43 


RCL 


435 


93 


• 




274 


43 


RCL 


328 


43 RCL 


382 


33 


33 


436 


00 







275 


40 


40 


329 


02 02 


383 


54 


) 


437 


02 


2 




276 


65 


X 


330 


52 EE 


384 


65 


X 


438 


00 







277 


43 


RCL 


331 


22 INV 


385 


53 


( 


439 


07 


7 




278 


27 


27 


332 


52 EE 


386 


43 


RCL 


440 


09 


9 




279 


55 


4 


333 


32 X:T 


387 


36 


36 


441 


65 


X 




280 


43 


RCL 


334 


43 RCL 


388 


75 


— 


442 


73 


RC* 




281 


28 


28 


335 


35 35 


389 


43 


RCL 


443 


00 


00 




282 


95 


= 


336 


52 EE 


390 


34 


34 


444 


45 


yx 




283 


42 


STO 


337 


22 INV 


391 


95 


= 


445 


01 


1 




284 


39 


39 


338 


52 EE 


392 


42 


STO 


446 


93 


• 




285 


94 


+/- 


339 


22 INV 


393 


36 


36 


447 


01 


1 




286 


85 


+ 


340 


58 FIX 


394 


61 


GTO 


448 


95 


= 




287 


43 


RCL 


341 


22 INV 


395 


12 


B 


449 


65 


X 




288 


40 


40 


342 


67 EQ 


396 


76 


LBL 


450 


73 


RC* 




289 


95 


= 


343 


47 CMS 


397 


19 


D' 


451 


00 


00 




290 


42 STO 


344 


22 INV 


398 


87 


IFF 


452 


65 


X 




291 


38 


38 


345 


86 STF 


399 


01 


01 


453 


93 


• 




292 


87 


IFF 


346 


00 00 


400 


52 


EE 


454 


00 







293 


00 


00 


347 


43 RCL 


401 


99 


PRT 


455 


08 


8 




294 


14 


D 


348 


36 36 


402 


75 


— 


456 


03 


3 




295 


43 


RCL 


349 


19 D' 


403 


73 


RC* 


457 


04 


4 




296 


40 


40 


350 


61 GTO 


404 


48 


48 


458 


65 


X 




297 


19 


D* 


351 


12 B 


405 


95 


= 


459 


92 


RTN 




298 


43 


RCL 


352 


76 LBL 


406 


55 


• 


460 


76 


LBL 




299 


42 


42 


353 


47 CMS 


407 


73 


RC* 


461 


11 


A 




300 


32 


X:T 


354 


48 EXC 


408 


48 


48 


462 


86 STF 




301 


43 


RCL 


355 


36 36 


409 


95 


= 


463 


01 


01 




302 


47 


47 


356 


48 EXC 


410 


33 


X2 


464 


42 


STO 




303 


22 


INV 


357 


34 34 


411 


44 


SUM 


465 


42 


42 





24 



StepeO-15 
Reads 

No. of 

IncHollres 

mottriol balance mode 



X 



LBLE 
Step* 16-23 
Reads 
Al203conc. 



T 



Steps 24-42 

Reods 

Circuit code 

Input slurry Al2 O3 wt. 

Input slurry volume 

Wosri woter volume 

Cake total liquid volume 



_£ 



LBL GTO 
Steps 43-63 

Coverts AlgQ, 
Cone to wt. 



LBL A 

Steps 64-96 

Reads 

Coke internal liq vol 

Voids shrink const. K 



I 



LBL B 
Steps 97-176 
Calculates f ,n,ond 
form cake AI2O3 



Step 177 

Print s 

Form cake AI2O3 



J i 1 L 

LBL D 
Steps 178-211 
Calculates 
Wash hq Al 2 3 



Steps 212-214 | , ' 

Checks balance flog 



Steps 215-217 

Prints 

Wash liq Al 2 3 



LBL sin 
Stepe 218-294 
Calculates f,n, and 
cake liq AI2O3 



Steps 295-305 

Prints 

Washed cake Al 2 3 



Steps 306-319 

Prints 

SSE 



1 



LBL C 

Steps 320-325 

Checks balance flog 



Steps 326-343 
Checks for system 
material balance 



LBL D' 




Steps 396-400 




Subroutine 




Checks predictor flag 


31 
| 


1 


1 




Steps 401-416 




Prints : 




Al 2 3 wt 




1 

a 


Sums (error)2 










' 








LBL EE 








Steps 417-419 






Return 







LBL B ' 

Steps 420-430 

Prints : 

Wash water Al 2 3 



LBL Y » 

Steps 431-459 
Subroutine 
Converts AI2 O3 
Cone, to wt. 



LBL A 


Steps 460-473 


Subroutine 


Initializes 


-No. of washes 


-Predictor mode 



balance flog set 



Steps 344-351 
Sets balance flag 



LBL CMS 
Steps 352-395 
Choose new 1st 
wash filtrate AI2O3 



FIGURE B-1.— Schwmtfc of TI-59 program for material balance using the shrinking voids model. 



25 



APPENDIX C.-MEASUREMENT OF LEACH RESIDUE POROSITIES AND DENSITIES 



The porosity of calcined kaolin clay that has been 
leached in a stoichiometrically 5-pct-excess aqueous 26 
wt-pct HCI was determined as follows: 

A sample of the leached solids (still saturated in the 
leach liquor) was first dried with a cloth dish towel to 
remove external moisture. Then a weighed amount, W s , of 
these solids was dropped into a tared 25-ml pycnometer 
which was then filled to the mark with leach liquor of 
known density, p|j qu0 r- Then using the weight, Wg, of leach 
liquor added, the volume, V s , of the original towel-dried 
solids was found to be 



V s = 25 



Pliquor 



milliliters. 



(C-1) 



The solids were then thoroughly washed with distilled 
water and dried in an oven at 125° C to obtain a dry solids 
weight, W ds . From this weight, the weight, Wj, and volume 
V|, of internal liquor (inside the particle pores) were found 
to be 



Wj = W s - Wds grams and 
Wj 



V, = 



milliliters. 



(C-2) 
(C-3) 



Pliquor 



This information is then used to find the percent poros- 
ity, P, by volume 



P = 100 



(C-4) 



V, 



Table C-1 summarizes these porosities along with the 
wet particle densities 



W s 
p s = — grams per milliliter, 

V s 



(C-5) 



and an absolute solids density 



W ds 
Pds = — grams per milliliter. (C-6) 

V ds 

TABLE C-1.— Summary of 1 -day-old leach residue 
densities and porosities (20° C) 





Wet 
particle 
density' 

g/cm 3 


Dry particle 

absolute 

density, 

g/cm 3 


Dry particle 
porosity, 
volume 
percent 


Feed size, mesh: 

Minus 10 plus 14 

Minus 14 plus 20 

Minus 20 plus 28 


1.687 
1.619 
1.728 


2.212 
2.146 
2.181 


56.24 
53.40 
54.14 


Average 


1.68 


2.18 


54.6 



Vliquor = 1-2776. 



26 



APPENDIX D.— CAKE LIQUOR DENSITY AND VOLUME DETERMINATIONS 



The original wet cake is weighed, W ws , and washed with 
a weight of wash water, W ww . The repulp liquor obtained 
has a density, p R , and weight-percent Al 2 3 , P R , as does 
the original liquor in the cake, density, p^, and weight- 
percent, P . The dry washed cake has a weight, W ds . 

The relationship between density and weight-percent 
Al 2 3 is 



p = 1 + 0.02079 (P ) 11 , 
p R = 1 + 0.02079 (Pr)1-1. 

The weight of the original liquor is 

w L = w ds + w ws . 

The weight of alumina in the original liquor is 

Pr 
W A = — (W L + W^). 
100 

The percent alumina in the original liquor is 

W A 



P = 100 



(D-1) 
(D-2) 



(D-3) 



(D-4) 



(D-5) 



Wi 



To obtain the original liquor density, P in equation D-5 
is substituted into equation D-1 



p = 1 + 0.02079 100 



W A V-1 
W 



(D-6) 



Substitution of the W L and W A values of equations D-3 
and D-4 gives 

/ /W ws - W ds + W WW \\1-1 

P = 1 + 0.02079 P R ] . (D-7) 

\ \ W ws - W ds // 



Solving equation D-2 for P R gives 



0.02079 



(D-8) 



Finally, substitution of this value into equation D-7 gives 

/pr - A 1/1:1 
p = 1 + 0.02079 



\0.02079j 



f W ws - W ds + W v 




1.1 



- w ds 

w ws - w ds + WwwV' 1 



(D-9) 



W ws - W ds 



With this density, the liquor volume per weight of cake 
solids must then be 



V, = (W ws - W ds )/ Po . 



(D-10) 



27 



APPENDIX E.-NOMENCLATURE 



A = Pounds Al 2 3 in liquor stream (usually subscripted). 

A f = Pounds Al 2 3 in wash water stream (value usually zero). 

C eq = Equilibrium AI2O3 concentration in liquor, pounds per gallon. 

Cf = Final AI2O3 concentration in liquor, pounds per gallon. 

Cj = Initial AI2O3 concentration in liquor, pounds per gallon. 

C w = Solute concentration in wash liquor, pounds per gallon. 

C-| = Solute concentration in feed cake liquor, pounds per gallon. 

C 2 = Solute concentration in washed cake liquor, pounds per gallon. 

e = Natural logarithm base, 2.71828. . . 

f = Fraction of cake AI2O3 removed during wash with salt-free wash liquor. 

j = Number of perfect mixing cells. 

k = Voids shrinkage constant, square gallons per pound. 

N = The wash ratio, Vi/V . 

R = Residual, a theoretical measure of solute remaining in cake after washing. 

tc = Time constant in diffusion model. 

V = Gallons of liquor (usually subscripted). 

W = Pounds of liquor in stream. 



Greek letters 



A = Change in the quantity that follows, 
p = Liquor density, pounds per gallon. 



Subscripts used with A and V 



e = External liquor fraction of cake particle, 

i = Internal liquor fraction of cake particle, 

t = Total liquor fraction of cake particle. 

= Feed cake. 

1 = Wash liquor. 

2 = Wash filtrate. 

3 = Washed cake. 

4 = New feed cake. 



•U.S. GOVERNMENT PRINTING OFFICE l 1982 0-367-'l 1 t8/626 



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